The biggest danger for British number one was the rain, as play was stopped twice, including moments after Norrie could not take the third match point that came his way deep into the third set.
He returned to get the job done quickly with a 6-0 7-6 (7-3) 6-3 victory, five hours after he first went to court.
The ninth seed will face another Spaniard, Jaume Munar, in his next match.
Norrie, who had never made it past the third round of a Grand Slam, made a confident start on track 2 and ensured an early break when Andujar put up a long lead.
A love affair was followed by another breather, while the Briton framed a volley at the net and watched it spin perfectly over the net, while his hands up in apology did little to improve the mood. of the Spaniard who was 4-0 behind. any time.
Andujar served to avoid a bagel, but could not do so as Norrie finished the set in 23 minutes, yielding nine winners and just one unforced error in a clinical performance.
A first flash of the morning came when Norrie was broken in the opening match of the second set, and it was the afternoon when he got a chance to fix it as the rain brought the proceedings to a halt.
Contrary to what was seen before the weather intervened, it was a very competitive set of tennis, perhaps aided by Andujar being liquidated for having to overplay a few points after Hawk-Eye revealed that two big hands had landed from him on the line with a matter of millimeters.
Norrie got the break to get things back on track and it went to a tie-break, where Andujar’s forehand let him down just at the wrong time. The Brit brought up three set points and needed just one.
The third set was a relatively routine affair, Norrie taking control in the third game when a breach of love was sealed with a double fault from Andujar.
After holding on to make it 5-3, Norrie saw three game points come and go on the Spaniard’s knockout, with play suspended as the game turned around for a sixth time.
A seventh and eighth two followed as the players stepped back to glorious sunshine, and Norrie was finally able to get the job done with a beautiful forehand winner along the line.